Computation of Certain Topological Indices of Nanotubes

Among topological connectivity indices, the atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are of vital importance. The ABC index is defined as: ABC(G) = Σ _uv∈E(G) √(d _G (u) + d _G (v) – 2)/(d _G(u)d _G (v)), while GA(G) index also defined as follows: GA(G) = Σ _uv∈E(G)((2√d _G (u)d _G (v))/d _G (u) + d _G (v))), where d _G (u) denotes the degree of vertex u ∈ V(G). Recently, the fourth version of ABC index is proposed by Ghorbani et al. defined as follows: ABC ₄(G) = Σ _uv∈E(G) √(δ _G (u) + δ _G (v) – 2/(δ _G (u)δ _G (v)). The fifth version of GA index is introduced by Graovac et al. defined as follows: GA ₅(G) = Σ _uv∈E(G) ((2√δ _G (u)δ _G (v))/(δ _G(u) + δ _G (v))) , where δ _u = Σ _uv∈E(G) d _G(v). In this paper, we study the fourth atom-bond connectivity index ABC ₄ and fifth geometric-arithmetic index GA ₅ and give close formulae of these indices for HAC ₅ C ₇[p, q], HAC ₅ C ₆ C ₇[p, q] and TUC ₄ C ₈(R)[p, q] nanotubes and their corresponding nanotori. We also give a characterization of k-regular graphs with respect to their fifth geometric-arithmetic index.